I'm not sure what it's supposed to be but I would lean towards saying it is the responsibility of the pilot.
ATC should provide separation from other IFR traffic but will not separate you from VFR traffic, or at least is not obliged to.
The other thing that comes to mind is at a slight tangent, but I find it fairly common to ask for a visual approach and for ATC to say 'do you have the A320 on a 6 mile final in sight?" and then "cleared visual approach behind the landing A320, recommended spacing is xnm" which puts the separation between two IFR flights on the pilot following
AFAIK.